ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106

LAGRANGIAN DESCRIPTION AND FINITE ELEMENT ANALYSIS OF RELUCTANCE ACCELERATOR CIRCUIT MODEL

Gökhan DINDIŞ1*

1Eskişehir Osmangazi Üniversitesi, Mühendislik Mimarlık Fakültesi, Elektrik Elektronik Mühendisliği Bölümü, Eskişehir, ORCID No : http://orcid.org/ 0000-0001-5642-7212

Keywords Abstract Lagrangian Description Lagrangian description of electrical circuit model of a reluctance type accelerator system Reluctance Accelerator is introduced. The effectiveness of the general purpose magnetostatic finite element Electromagnetic analysis (FEA) tools on the introduced model is demonstrated. Electrical equivalent Accelerator circuit model of the system lays out many properties of the system in terms of voltage, Coil Gun current, power or energy distributions of electrical components, and also electrical Electromagnetic Launcher equivalent of the mechanical components, in an easily perceivable form. These type of actuators are simple mechanisms when they are in steady state. However, when they are in transient, their dynamics may become very complex. The aim of the study is to show how this approach allows many researchers without the full electromagnetic background, to model and understand the dynamics of the reluctance type linear or rotary motor or actuators whether they are saturated or not. To exhibit the validity of the proposed model and solution of dynamic behaviors with FEA, it is verified on a basic capacitor discharge type driver circuit including the electromagnetic accelerator coil and projectile.

RELÜKTANS TÜRÜ İVMELENDİRİCİ DEVRE MODELİ İÇİN LAGRANGE TANIMLAMASI VE SONLU ELEMAN ANALİZLERİ Anahtar Kelimeler Öz Lagrange Tanımlama Relüktans türü ivmelendirici devre modeli için uygun bir Lagrange tanımı sunulmuştur. Relüktans Türü İvmelendirici Magnetostatik sonlu eleman analizi yapan genel amaçlı programların, sunulan model Elektromanyetik üzerinde ne kadar etkin bir şekilde kullanılabileceği gösterilmiştir. Bir sistemin elektrik İvmelendirici devre modeli, sistemi oluşturan parçaların davranışlarını, mekanik parçaların elektriksel Bobin Silahı eşdeğerleri de dahil olmak üzere akım, gerilim, güç ve enerji dağılımı gibi karşılıklarıyla Elektromanyetik Fırlatıcı kolay algılanabilir bir şekilde ortaya sermektedir. Bu tür tahrik sistemleri durağan çalışma şartlarında oldukça basit dinamiğe sahip olmakla birlikte geçiş durumlarında oldukça komplex davranış gösterirler ve çalışma dinamiklerinin anlaşılmaları güçtür. Çalışma, böyle bir yaklaşımın tam bir elektromanyetik alt yapıya sahip olmayan araştırmacılara doymuş veya doymamış relüktans tipi doğrusal veya dairesel hareket eden motor veya elektrikli tahrik sistemlerinin ve dinamik yapılarının daha iyi anlaşılabilmesi için yapılmıştır. Önerilen model ve dinamik davranışlarının sonlu eleman analizleri ile çözümü, bir relüktans ivmelendirici düzeneği gerçeklenip kondansatör boşaltmalı bir devre ile sürülerek doğrulanmıştır. Araştırma Makalesi Research Article Başvuru Tarihi : 15.01.2020 Submission Date : 15.01.2020 Kabul Tarihi : 03.04.2020 Accepted Date : 03.04.2020

it possible what was unimaginable previously. Yet, it 1. Introduction seems there are fundamental issues still unknown to Using electromagnetic accelerators (EMA) to launch many researchers. Thanks to the newly developed projectiles at high velocities has attracted researchers affordable hardware and software tools so that more for more than a century. With the continuously and more newcomers with different perspectives can emerging technologies, significant number of studies step into this specific field and contribute even more. have contributed to the topic and have led to The well known survey paper on Electromagnetic successively greater progress (McNab, 1999). Recent Momentum (Griffiths, 2012) would be a good start point electronic and computational advancements have made for detailed exploration for any type of magnetic

* Sorumlu yazar; e-posta : [email protected]

ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106 accelerators. it is focused specifically on the reluctance type accelerators. EMAs may be classified under two categories. One of the categories can be described as railgun type accelerators Before going into more detailed research a well defined which is not in the scope of this research. The second model for the system is essential. There is a well known category on the other hand is coil gun type accelerators and established unified physical model for the which has two subcategories: reluctance type and reluctance type accelerators in the literature (Slade, induction type. Both subcategories have some common 2005). A simplified physical model is developed from features such as; one or more excitation coils located on basic principles and cast in a Lagrangian form. However, a tubular runway or a barrel, and accelerating a in the model a longer armature than the coil has been projectile in the barrel to a high velocity with the thrust taken into consideration and includes many simplifying caused only by a without any electrical assumptions. Some of those assumptions obviously fail contact. “Induction type accelerators” is not a subject to with complex geometries, for example when the scope of this research, however these accelerators are geometry has a bottleneck section. Another article from far better than the reluctance type ones in regards of the same author introduces a fast finite element solver muzzle speed because practically no magnetic for a reluctance type accelerator which involves the saturation problems they face as reluctance type ones writing the FEA program itself as a solver (Slade, 2006). do. An induction type EMA uses a non-ferrous but Eddy currents and driving circuit elements are included conductive sleeve type projectile. Thrust forces for this in the model. However, for the more complex type of accelerators are based on the induction currents geometries it may not be a simple task for a researcher and Lorenz forces. Operational requirements of the without the strong electromagnetic background, to induction types differ from the reluctance types and write an electromagnetic finite element solver. A user very high speeds (e.g. 2.5 km/s) are possible (Kaye, friendly general purpose FEA program for magneto- 2005). Also, some design procedures available in the static problems on the other hand, makes the process literature which discusses for specific requirements of a much simpler. So that, as proposed in this manuscript, a super velocity launcher with 8 km/s muzzle speed well defined electrical circuit equivalent model using a (Balikci, Zabar, Birenbaum, and Czarkowski, 2007). general purpose FEA toolset seems to be another way to go. There are articles and conference papers found in A reluctance type EMA uses a low reluctance projectile the literature look somehow similar to one proposed or sometimes called armature or slug that is energized here. Some of them also use a general purpose FEA as a by the magnetic field generated by one or more driver solver. However, simulations are far from close coils. For the reluctance type accelerators, the force that approximations since most of them don’t even have accelerates the projectile can be explained by Maxwell projectile dynamics reflected on the electrical circuit stress tensors. The quantitative calculation of the models (Holzgrafe, Lintz, Eyre, and Patterson, 2012; magnetic forces for any geometry and any projectile Klimas,Grabowski, and Piaskowy, 2016; Khandekar, position can be done either by surface integral, the 2016). volume integral, or equivalent surface methods. Errors depend on the quality of the FEA solution but the volume integration methods are always more precise than the 2. Materials and Methods surface ones or the equivalent source ones (De 2.1. Driver Coil Medeiros, Reyne, and Meunier, 1998). Analytical solution is very difficult and needs a lot of A driver coil is the essential part of an EMA. One, approximations but FEA tools yield reasonable accurate focusing on a single driver coil and modeling its physical solutions for majority of cases. Limitations for the behaviors on either stationary or moving projectiles reluctance type accelerators seems still not well defined obviously will encounter a multi-stage coil cases too. In so far, and literature indicates much more research fact, when a projectile comes to the first stage with a work can be done. One may say that simple search yields non-zero speed its behavior is not different than that in answers for these limitations easily. However, over the following stages. Thus, examining a simple single hundreds of articles related to reluctance type stage driver coil is fundamental in understanding the accelerators, there was none that indicating a maximum mechanics of the operation. Copper is used generally for reachable speed data with a physical verification. the coil material because of its good electrical Literature indicates that induction and saturations are conductivity properties. Besides, its wide industrial important factors for determining limitations of usage makes it low cost and available almost anytime. reluctance types, and directs the speed or energy Windings should have a good insulation wire to wire and seeking researchers to go for induction types. That layer to layer. Stranded copper may add to the material might be the reason most of the researchers left the field cost but it is good for high frequency responses which without setting the approximated borders for key specially needed at very high velocities. An efficient coil parameters like speed, energy or efficiency. That was winding should have a high 퐿/푅 ratio, unnecessary air the main motive to start for this research and therefore pockets among the conductor wires should be avoided.

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On the other hand, high 퐿/푅 ratio makes the response the projectile material is preferred due to minimizing time of the system very slow for the intended sudden the eddy currents. Soft ferrites made out of sintered development of the drive current and it becomes more powders contain iron oxides combined with nickel, zinc, difficult to drive the coil under low voltages and in a and/or manganese have high resistivity against the tight time frame. The for a coil can be eddy currents. However their permeability is calculated as: considerably lower than solid or stacked ferrous materials; that’s why they perform poorly (Slade, 2005).

Experiences indicate that, for the low speed coil stages 푁2 퐿 = (1) mainly low permeability and better magnetic saturation ℛ specs are required. Geometry of the projectile is also important. Length for where ℛ is the total reluctance for the effected magnetic example, can be optimized to maximize the energy or field. Relative permeability properties of the copper, air, speed of the projectile (Daldaban and Sarı, 2016). fiber-glass, and plastic materials have almost the same as the vacuum has. Thus, if these are the main materials used for the coil and surroundings, the total reluctance 2.3. Driving Circuit is almost firm. When there is no saturation, for a given For the coil driving generally voltage-source type power geometry, 퐿 strictly depends on square of number of supplies are used. Obviously, they should have high turns 푁. Inside the conductor, L changes almost linear current capabilities and fast response times for the with 푁, but it has generally very small effect on the purpose. This can be done easily by connecting large resulting 퐿. To wind the same geometry with the same capacitor banks parallel to the output of the power type of conductor; if the number of turns N needs to be supply. That makes the impedance of the output of the increased, wire diameter should be decreased to fit into power supply, in contrast to the coil impedance, much the volume, or vice versa. If the diameter of the wire lower. Switch internal resistors also should be very decreases its ohmic resistance 푅 increases as L does. As small, and for the same reason hook-up wires should be a result, time constant 퐿/푅 stays almost the same. kept short and thick. However, changing the air/conductor ratio by winding with some air gaps is a different issue. Introducing a Silicon controlled rectifiers (SCR) are commonly used material with a lower reluctance into the field also for energizing the coils. Their high current and voltage effects the overall permeability by increasing the ratings and low internal resistivity makes them good inductance, and the 퐿/푅 ratio accordingly. This happens choices for this purpose. Capacitor or capacitor-bank exactly when a ferrous projectile breeches the coil. types discharge circuits are suitable for the operation. A simple circuit example can be given as in Figure 1 (a). The capacitor should have the right energy to dump to 2.2. Projectile the coil, so there would be no suck back effect beyond the center position. Because a standard SCR naturally Magnetic coupling between the coil and the projectile will not turn off until the current reaches zero. A gate should be maximized for a good power transfer. turn off thyristor (GTO) can be a solution (Rashid, 2011), Therefore, coil design and projectile design has to match but going into that direction is not much different than each other in sizes, in weights, and so on. For the using MOSFET or IGBT. In normal operation, when the reluctance type EMAs the projectile material should projectile is in the pull zone and SCR is set on, current have good magnetic permeability. Magnetic saturation starts to accelerate the projectile toward the center of reduces the magnetic permeability of the material at the coil. The total electrical energy left in the capacitor elevated current values, and therefore it should be and the coil itself should be equal to zero as soon as the avoided as much as possible for the overall efficiency. A projectile reaches the center. Otherwise, current will be simple projectile can be made from a piece of soft iron still flowing, and the projectile will start to decelerate. or ferrous steel, which is cylindrically shaped to fit the bore of the barrel with a smallest air gap. However this kind of material will have its own limitation due to high eddy currents. Eddy currents have negative effects on projectile’s speed and efficiency. Lorentz forces created by eddy currents create force in the negative direction when projectile is supposed to be pulled towards the center of the coil. If the material’s ohmic resistance is low, then the force will be stronger and will last longer. If the ohmic resistance is moderate, it will allow eddy currents for a shorter time frame by converting the power to the ohmic losses. High electrical resistance of 96 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106

2.4. Principles of Operation for the Reluctance Type Accelerator The physical principle of the operation for the reluctance type accelerators can be explained in simple terms. A magnetic field is a force field for a low reluctance material. The material has tendency to move into the direction of lowering the reluctance of the magnetic field. Considering this, following actions are taken to accelerate the projectile to the desired speed. A temporary magnetic field is generated in the coil by applying a current through the coil. Naturally, the projectile is forced to move to the center of the coil, since (a) there is potential for lowering the reluctance in that direction. It accelerates until the center point where reluctance is the lowest and speed is the maximum. After this point current should be suddenly cut off and temporary magnetic force field should be vanished as soon as possible. Otherwise the force field is going to pull the projectile back to the center. After switching the current off, a small, short lasting suck back effect might be permitted, since its decelerating effect is minimal and making it completely disappearing is a challenging task.

2.5. The Lagrangian Model

A simplified electrical circuit model for a reluctance type (b) accelerator is given in Figure 2. The voltage source Figure 1. Simple SCR (a) and MOSFET (b) Controlled stands for the capacitor bank used to drive the circuit. The total equivalent resistance R includes capacitor’s Circuits for Capacitor Discharge Type Coil Driving internal resistance, switch-on resistance and coil resistance all combined. The inductance 퐿(푥, 퐼) represents inductance of the coil which changes as a Metal oxide silicon field effect transistors (MOSFET) or function of projectile position and current flowing isolated gate bipolar transistors (IGBT) are also through the coil. Because of the high ampere-turn ratios commonly used for energizing the coils. We will use the projectile goes to magnetic saturation and it is obvious term transistor for both type of switches in this section. for 퐿 to be function of 퐼. The last part in the model Their high current, voltage, and low internal resistance represents the effect of the projectile’s mechanical values cannot not compete with the SCRs’, however properties. The projectile is magnetically coupled to the their high switching-on and switching-off properties circuit and its effect on the circuit will change give users a better control than a SCR can offer. In a significantly by its dynamic and physical parameters. simple use, when the transistor is turned off while some Magnetic force 퐹(푥, 퐼) applied to the projectile changes current is still passing through, a fly-wheel diode D2 and also by its position in the coil due to the degree of the a quenching resistor may handle the turn-off inductive magnetic coupling. On the other hand a moving high voltage spike and protect the transistor from projectile changes the magnetic reluctance and getting damaged (Figure 1 (b)). A snubber circuit therefore changes the in the magnetic parallel to the transistor switch is also advised if long circuit as well. It is well known that according to the hook-up wires used. In transistor controlled systems Faraday’s law, changing flux over the time induces one big capacitor-bank can be used for repetitive shots voltage on the coil it passes through. This phenomenon or multi coil driving purposes. Diode D1 is for the the taking place is the generator effect and it is observed in purpose of preventing electrolytic capacitors from any kind of motor or electro-mechanical actuator charging backward which may harm the capacitors in motion. That’s why a counter electromotive force (or the time thereafter. back-emf) should exist in the electrical circuit models too.

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expressed separately as in Equations (8)-(9), for the electromotive forces (emf) and mechanical forces respectively.

푑 휕ℒ 휕ℒ ( ) − + 푅푞̇ = 푈 (8) 푑푡 휕푞̇ 휕푞 푝

1 푑 휕( 푚푥̇2) ( 2 ) = 퐹(푞̇, 푥) (9) 푑푡 휕푥̇

where 푈푝 is the electromotive force field effect at the Figure 2. Electrical Circuit Model magnetic coupling of the moving projectile that is seen at the electrical circuit side and R represents the total circuit resistance. Then, Equation (8)-(9) yields Equations (10)-(11). There will be some induction currents on a solid iron or steel projectile. It is at neglectable levels for low slew rate current applications, but naturally rises up with the 푞 휕퐿(푞̇ ,푥) 푑퐿(푞̇ ,푥) − + 퐿(푞̇, 푥)푞̈ + 푞̇푞̈ + 푞̇ high slew rate current pulses required in the following 퐶 휕푞̇ 푑푡 coil stages. This induction will cause large Lorentz forces 푑 휕퐿(푞̇ ,푥) in negative direction during the breeching phases of the + 1 푞̇ 2 + 푅푞̇ = 푈 (10) 2 푑푡 휕푞̇ 푝 projectile. So, modeling for the high speed coil stages will be another challenging problem and it is not in the 푚푥̈ = 퐹(푞̇, 푥) (11) scope of this manuscript. Projectiles subject to fit the model proposed in this paper are assumed either running at low speed applications (less than about 50 When induction, hysteresis, and friction losses are all in m/s) or they have geometries to reduce the induction neglectable amounts, conservation of the power for the currents as in slitted solid projectiles (Barrera and electromagnetic coupling yields Equation (12). Beard, 2014) or projectiles made from laminated sheets or wire/epoxy cores (Slade, 2005). 푈 푞̇ + 퐹(푞̇, 푥)푥̇ = 0 (12) The Lagrangian of an electromechanical system ℒ , in a 푝 general sense, takes the form as in Equations (2)–(5) (Slade, 2005; Goldstein, Poole, and Safko, 2002). In plain words, whatever the power amount electromotive force supplies to the coupling has to be received by the mechanical system and should be turned ℒ = 풯 − 풱 (2) into mechanical power and vice versa. The equations are 풯 = 푇퐿 + 푇푚 (3) unified as shown in Equation (13).

풱 = 푉퐶 (4) 푞 휕퐿(푞̇ ,푥) 푑퐿(푞̇ ,푥) − + 퐿(푞̇, 푥)푞̈ + 푞̇푞̈ + 푞̇ 퐶 휕푞̇ 푑푡

Where 푇퐿 and 푇푚 are kinetic energy of the inductor and 푑 휕퐿(푞̇ ,푥) 1 + 1 푞̇ 2 + 푅푞̇ + 퐹(푞̇, 푥)푥̇ = 0 (13) projectile respectively, 푉퐶 is the potential energy of the 2 푑푡 휕푞̇ 푞̇ capacitor. In expanded form they are shown in Equations (5)-(7). 푑퐿(푞̇ ,푥) 푑퐿(퐼,푥) Finally, when 푞̇ and are replaced by 퐼 and 푥̇ 푑푡 푑푥 1 ( ) 2 respectively, the equation becomes: 푇퐿 = 2퐿 푞̇, 푥̇ 푞̇ (5) 1 2 푇푚 = 2푚푥̇ (6) 푑퐼 휕퐿(퐼,푥) 푑퐼 푑퐿(퐼,푥) 푞2 −푈 + 퐿(퐼, 푥) + 퐼 + 퐼푥̇ 푉 = 1 (7) 푐 푑푡 휕퐼 푑푡 푑푥 퐶 2 퐶 푑 휕퐿(퐼,푥) 퐹(퐼,푥) +1 퐼2 + 푅퐼 + 푥̇ = 0 (14) 2 푑푡 휕퐼 퐼 where 푚 and 푥 are mass and position of the projectile, and 푞 is the charge of the capacitor. Then, including the Equation (14) clearly lays out the important properties dissipative term 푅푥̇, equations of motion can be of the system. First of all, the Equation 14 is nothing 98 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106

휕퐿(퐼,푥) different than the Kirchhoff’s voltage law. The in 2.6. Finite Element Analyses 휕퐼 the equation appears to be representing the magnetic The finite element analysis is basically a numerical saturation effect and it is neglectable at currents of non- method for finding approximate solution to boundary 퐹(퐼,푥) saturating levels. The term 푥̇ which may be value problems in partial differential equations. It 퐼 퐹(퐼,푥) subdivides a large problem into smaller, simpler parts expessed as 퐼푥̇ as well, causes some of the back-emf 퐼2 that are called finite elements. When these finite voltage which is a natural response of the projectile elements define the model of the entire system behavior moving in the coil. Some back-emf voltage is also well enough, assembled simple solutions of these small, 푑퐿(퐼,푥) produced by its co-energy counterpart 퐼푥̇. Until simple elements will yield approximate solution for the 푑푥 퐹(퐼,푥) entire model of the system. When elements are chosen the saturation point, ratio follows a constant curve 퐼2 much smaller, naturally approximations gets more and it gets smaller with the increasing saturation level. accurate but computation time gets enormously longer. Figure 5 (a) and (b) in section 4.2 demonstrate this illustratively. Note that, if we denote 2.7. Initial Analyses with FEMM: Inductance and

Force Responses of the Coil with a Projectile 퐹(퐼,푥) 퐾 = (15) 푚 퐼2 In this article a public licensed “FEMM” software is used which gives simple planar or axisymmetric solutions for

magneto static problems. There are many research 퐹(퐼,푥) examples in the literature using this software as their then, the term 퐼푥̇ becomes equal to 퐾푚퐼푥̇ for the 퐼2 main tool (Klimas at al., 2016; Baltzis, 2010). It offers smaller linearized sections, and it is proportional with solutions for planar and axisymmetric problems the coil current or projectile speed. This is a typical DC (Meeker, 2004). A simple axisymmetric geometry of a motor/generator behaviour. When the speed is constant coil and projectile can be easily drawn and solutions can more excitation current produces more voltage, and be calculated as in Figure 3. Geometry is drawn in cross when excitation current is constant more speed sectional form. Since it is cylindrically rotated around produces more back emf voltage; which is not 휕퐿(퐼,푥) 푑퐼 the central axis, drawing only one half of the cross surprising. The other terms in Equation (15), 퐼 휕퐼 푑푡 section of the geometry will be just enough. Results can 푑 휕퐿(퐼,푥) and 1 퐼2 appear to be representing the electro- be displayed or exported in graphical or numerical form. 2 푑푡 휕퐼 It has a build-in library for many magnetic and motive force introduced by the saturating effect of the conductive circuit materials. If the properties of the magnetic field. 푈 in Figure 2 represents emf responses 푚 materials are chosen or edited correctly, even with of the terms related to the moving projectile. In a motion default mesh size, errors are generally in single digit of acceleration with a proper current flow; projectile percents or less. When the system has non-symmetric builds more kinetic energy while speeding up, but coil structure, then, any of those 3D FEA software packages builds more magnetic energy too as its inductance also can be used for the solutions. increases. Naturally, as power consumers, during this kind of process they both put their responses to the Solving the Lagrange equations requires solving of circuit as electro-motive force effects. If all the equations 퐹(퐼, 푥) and 퐿(퐼, 푥) data for any given current value at any are re-organized we get the differential equations given projectile position. However, before jumping below: straight to the solution of the dynamic problem, doing some static analyses is very helpful. This gives us some

sense about the behavior of the system. In any FEA 푑퐿(퐼,푥) 푑 휕퐿(퐼,푥) 퐹(퐼,푥) 푈 − 퐼푥̇−1 퐼2−푅퐼− 푥̇ 푑퐼 푐 푑푥 2푑푡 휕퐼 퐼 suitable for this purpose, the geometry can be drawn = 휕퐿(퐼,푥) (16) 푑푡 퐿(퐼,푥)+ 퐼 and simulated calculations can be recorded. In FEMM, 휕퐼 the geometry is animated step by step, or mm by mm, 푑푈 1 푐 = − (17) 푑푡 퐶 and results can be recorded automatically via a companion script language tool, called Lua . The 푑2푥 1 = 퐹(퐼, 푥) (18) procedural steps are itemized below for obtaining the 푑푡2 푚 finite number of 퐹(퐼, 푥) and 퐿(퐼, 푥) data for the range of interest. These differential equations (Equations (16)-(18)) can be solved by numerical analysis methods iteratively for given 퐹(퐼, 푥) and 퐿(퐼, 푥) values. An FEA tool comes handy to obtain these values, since it gives us magneto- static solutions for any given current and position values. Section 4 describes how exactly it can be done.

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center, in this type of accelerators the main force direction is always toward to the center of the coil (Figure 4 (b)). Thus, when accelerated projectile passes the center the force becomes negative (slows the armature down). Energy will be transferred from the armature back into the magnetic field.

Figure 3. Axisymmetric FEA Model of Coil with Entering Projectile (left) and Field Solutions

FEMM Procedure to obtain the current responses

 Define the materials to be used. Import from libraries if they are already defined, and modify if (a) needed  Draw the geometries for initial position  Define the boundaries  Write a code in Lua script language for iterating the process N times for ; . analyzing and displaying the output fields, . calculating “Weighted Maxwel Stress Tensor Force” value, . calculating equivalent “Inductance” value, recording the values on a file, . advancing projectile 1mm further towards the muzzle direction

 Execute the Lua code (b) Results verify that when projectile is at the entrance the Figure 4. Typical Inductance (a) and Force (b) Results inductance is minimum and when it is right at the center via FEMM Analyses it reaches the maximum. Inductance change per distance at a constant current level appears to be determining the force, which also makes sense. When Because of the magnetic saturation properties of the this ratio is at the maximum the force is also at the projectile, its B-H curve will be almost linear up to a maximum. If the ratio is zero then so is the force. In fact, certain current level, but after that point, it will start to force becomes zero in two cases; the first case is when act nonlinearly. Therefore, more FEMM analyses have to the projectile is right in the middle and the second is if be done with different current values until and after the the projectile is at the infinity. Therefore, we see non- saturation, so that the effects become visible and the zero values at both entrances as shown in FEMM behavior of the system due to saturation can be analysis graphs in Figure 4 (a). The reason is that the observed and commented. reluctance of the projectile at the entrance still effects the total reluctance in some small amounts. One more The graphical results for the data in Table 1, for the coil- detail needs to be mentioned; that is the direction of the projectile setup, are shown in Figure 6 (a) and Figure 6 force. Since the reluctance becomes lower towards the (b). These comparative figures demonstrate the

100 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106 saturation effects on forces for different current values. Table 1. Parameters Used in Simulation Figure 6 (a) shows the force graphs for different current variable value Units values. More force levels for higher currents are still C 0.0705 Farads possible after the saturation. But the force levels will not R1 0.015 ohms be proportional with square of the current I anymore. It RL 0.11 ohms is clearly seen in Figure 6 (b) that normalized force N (number of turns) 125 turns graphics for 1000 amp-turns and 2500 amp-turns are d1 (internal dia. of the bobbin) 21 mm almost identical at unsaturated current levels. I1 (internal length of the 48 mm Furthermore, at saturated current levels, increasing the bobbin)a1 (wire diameter) 21 mm current starts to move the peaks of the curves from the R2 0.25 ohms center toward the entrance of the coil. The tools used in R3 0.0025 ohms regular FEMM simulations are made for magneto-static Rs (switch and hook-up wire 0.01 ohms problem solving, therefore induction or eddy currents, res)l2 (projectile length) 50 mm hysteresis effects, skin effects etc. are not accounted. d2 (projectile diameter) 10 mm These effects are generally not in considerable amounts m (mass) 30 grams Uc (initial cap. voltage) 50 volts at low speeds and will cause minor differences in the Ii (initial inductance current) 0 amperes results. x (initial position) -48 mm v (initial velocity, 푥̇) 0 m/s

2.8. Simulation of Lagrangian Model with the Integrated FEMM Software For the verification purpose of the proposed Lagrangian model of the system, the simple circuit model in Figure 2 is taken into consideration. This made experimental setup easy to implement too. Parameters are given in Table 1. For the current sensing four parallel connected 0.01 ohm resistors are used. Five paralleled IRFP064 MOSFETs used as a switch in the experimental circuit and catalog on-resistance value for each one shows 9 milliohms, if they are driven properly. So, equivalent switch resistance makes about less than 2 milliohms. In FEMM, by using the companion scripting language (a) Lua, it is straightforward to collect 퐹(퐼, 푥) and 퐿(퐼, 푥) data with given 퐼, and 푥 values. So, analyzing the coil behaviors under saturation conditions becomes possible by the simulation. The Lua language is a “MATLAB” like scripting language. Solving the differential equations of the model is possible iteratively, by solving via FEA ‘on the go’. The following procedure explains how it can be done: FEMM Procedure for Solving the Equations of the Motion: For the magnetic circuit;  Define the materials to be used. Import from libraries if they are already defined, and modify if needed

(b)  Draw the geometries for initial position Figure 5. Force Data (a) and Normalized Force Data (b)  Define the boundaries of FEMM Results For Lagrangian model;  Define the parameters capacitance C, resistance R, projectile mass m 101 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106

 Define the initial conditions for , I, x, ẋ  Start with time t = 0 (time frame k = 0) To iterate for the desired time period; • Solve magnetic circuit, obtain 퐹(퐼, 푥) and 퐿(퐼, 푥) • Solve magnetic circuit again, this time for the partial derivatives via small variations

• Solve differential equation, obtain k th time frame values • Save k th time frame values • Update the position x and current value I in the magnetic circuit for the next time frame (b) • Update time t, and time frame k (t ← t +∆ , k ← k +1) Figure 6. Component Voltages • Repeat iteration until reaching the desired condition The outputs are rewarding. From the electrical circuit model point of the view, graphical illustrations clearly 3. Case Studies and Findings lays out dynamic behavior of the reluctance type 3.1. Case Study 1: Simple Capacitor Discharge accelerator ( Figure 6 (a) and (b)). Curve “aa” shows the capacitor voltage change during the discharge. It is To evaluate the behaviour of the dynamic system for a apparent that, each component of the circuit model of simple capacitor discharge, a Lua script file is prepared the coil reacts with its own unique dynamic behavior. and executed according to the procedure given to solve Inductive component “bb” and resistive component “cc” the equations. The results are recorded in a text file and of the coil clearly dominate among the rest and drawn by using MATLAB graphical tools. The iteration demonstrate the largest voltage swings. Note that started at the position -48mm and continued for 10 resistive component voltage changes linearly with the milliseconds with 0.1 millisecond time steps. current and has a peak around t = 2.6ms. After that peak, when current value starts to decrease, inductive component starts to act as generator and try to keep current steady. That’s why its voltage changes the polarity and continues that way until current starts to increase again around t=7 ms. At this point the projectile is basically just after the middle point and because of the suck back effect of the coil it gives some of its kinetic energy back to the system. It is nice to observe that, some of the energy shows itself as increasing current effect. Of course increasing the current means also increasing the energy of the inductive component. Obviously some big portion of the energy becomes heat at the resistive component. When we look at the curve “dd”, we see the moving projectile’s responses. This response is called as “back electromotive force” of the projectile just like in the rotary DC motors. As it speeds up, more voltage drops will be seen. However, close to (a) the middle of the coil, there is not much magnetic potential differences, so we lose the emf effect of the projectile. Beyond the middle point the direction changes the other way around, and we see the projectile as an energy source. When it gives energy back to the system, sometimes the contribution may not be enough to increase the current but at least slows the current reduction rate. In the accelerator this is not a desired situation, since the main goal is to keep increasing the 102 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106 kinetic energy or the speed of the projectile. The curve 3.2. Case Study 2: Sensor Controlled Capacitor “ee” on Figure 6 (b) shows the total coil voltage which Discharge could be measured from both ends of the physical coil. As a second case study, simulations are done for the Some voltage drops will be in the internal resistors of controlled discharge situations. Assumed that discharge the capacitors and switch components, and that effect is event started by turning the switch on when the also seen as curve “ff”. projectile is at the entrance, and switch off when the Force and velocity graphics of the results are given in projectile is reached the center. Formulation of the Figure 7. After the middle point (around t = 7.2 ms), Lagrange model slightly changed when the control force direction changes and projectile start to lose its switch is off; capacitor is taken out of the equation and speed. It is clearly seen that force is very high around t extra resistor is added in series with an ideal fly-back = 4 ms. Since some of the energy is gone for the resistive diode. Two sub-cases are evaluated; one is with a zero losses, suck back effect around t = 8.7 ms is not high as ohm resistor value, and the other one is with a proper it was at the beginning. As a result, projectile will still value to act like a quenching resistor. Since the current have some speed while exiting the force field. The rates might be quite high, we can not choose a big value simulation results of the model reveals a good to create excessive voltage spikes. The one with fly-back supporting information. diode alone (Figure 8(a)) demonstrated a little higher suck back effect than the other one (Figure 8 (b)), just as

expected. Scientific and publication ethics guidelines are followed in all phases of this study.

3.3. Experimental Work For the verification of the proposed model a simple apparatus has been prepared mechanically as shown in Figure 9. System has an acrylic barrel as a runway. The coil specifications are same as those used in the simulation so that the comparisons make a quite sense. Position sensor made out of an infrared LED and photo- transistor pair and used for detecting when the projectile is at the center. Since it is located at the entrance and projectile length is equal to the coil length, as soon as projectile passes the sensor the switch can be (a) turned off. Speed sensors are made out of two optical sensors placed 10 mm apart and located after the coil. A micro-controller is used for controlling of the entire operation. The capacitor bank is charged to 50V, the switch is turned on by pushing the fire button to start discharging the capacitor bank into the coil, and let the current flowing until the projectile reaches the coil center. Then, while projectile is exiting the coil the muzzle speed is measured by using the time delay between two sensor signals. All experiments are repeated numerous times. Coil voltage and current values are recorded by using a storage oscilloscope. The muzzle velocity values are also recorded for each experiment.

(b) Figure 7. Force (a) and Velocity (b) Outputs

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signals show exactly the same behavior. The muzzle speeds are 10.2 m/s and 9.1 m/sn (averaged) obtained by the simulation and real measurements respectively. Figure 11 and Figure 12 shows sensor controlled cases of the experiment results and the simulation results. The velocity obtained at case 2b is again slightly higher than the velocity obtained at case 2a as expected. The simulated velocity results for the case 2a and 2b are 14.4 m/s and 15.3 m/s, and measured ones are averaged around 12.2 m/s and 13.1 m/s, respectively. There is a very important point here to be mentioned; since the solid projectile acts like a ’s secondary by itself, induction occurs for high dI/dt rates even in the armature itself. Therefore, quenching the primary circuit (the coil itself) does not take the energy of the (a) magnetic field right away. Note that, in the proposed model induction currents are not taken into account. The nonlinear semiconductor effects during the quenching time is not taken into account either.

(b)

Figure 8. Velocity Graphics for Fly-back Diode Alone (a) and with Quenching Resistor (b) (a)

Figure 9. Single Stage Experimental Setup (a) and Used Projectile (b)

4. Discussions Results are very similar to the ones obtained through (b) the simulations. Figure 10 (a) and (b) show the current Figure 10. Measurement and Simulation Results for and voltage values. Magnitudes of simulation coil currents and muzzle velocities are close but slightly Case 1 higher than experimental ones. Waveforms of the

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(a)

(b) Figure 12. Measurement and Simulation Results for Case 2b

5. Conclusions A well suited Lagrangian derived electrical circuit model for a reluctance type magnetic accelerator is introduced. With the electrical circuit model, each component’s behavior can be studied individually to understand the whole. For the geometries with more complexity, using a user friendly general purpose FEA program for magnetostatic problems makes this kind of problems easy to deal with. Proposed model with the use of (b) general purpose FEA tool might be another preferred Figure 11. Measurement and Simulation Results for way to go for the engineers without a strong electromagnetic background. When the parameters are Case 2a entered correctly, model offers accurate equations of the motion. A single stage accelerator is evaluated by simulations and a physical system setup. Results obtained by the simulations and the actual experiments verify the effectiveness of proposed model and toolset for the low speed reluctance type accelerators.

Conflict of Interest There is no conflict of interest.

References Balikci, A., Zabar, Z., Birenbaum, L. & Czarkowski, D. (a) (2007). On the design of for super-velocity launchers. IEEE Transactions on Magnetics, 43(1), 107-110. doi: http://doi.org/10.1109/ TMAG.2006.887651 Baltzis, K. B. (2010). The finite element method magnetics (FEMM) freeware package: may it serve as an educational tool in teaching electromagnetics?. Education and Information Technologies, 15, 19-36. doi: https://doi.org/10.1007/s10639-008-9082-8 105 ESOGÜ Müh Mim Fak Derg. 2020, 28(2), 94-106 J ESOGU Engin Arch Fac. 2020, 28(2), 94-106

Barrera, T. & Beard, R. (2014). Exploration and Magnetics, 42(9), 2184-2192. doi: verification analysis of a linear reluctance https://doi.org/10.1109/TMAG.2006.880395 accelerator. 17th International Symposium on

Electromagnetic Launch Technology (EML), La Jolla, CA, USA. doi: https://doi.org/10.1109/EML Daldaban, F. & Sarı, V. (2016). The optimization of a projectile from a three-coil reluctance launcher.

Turkish Journal of Electrical Engineering & Computer Sciences, 24, 2771-2788. doi: https://doi.org/10.3906/elk-1404-18

Griffiths, D. J. (2012). Resource letter em-1:

electromagnetic momentum. American Journal of Physics, 80(1), 7-18. doi: https://doi.org/10.1119/ 1.3641979

Goldstein, H., Poole, C. P. & Safko, J. L. (2002). Classical mechanics. Pearson. Holzgrafe, J., Lintz, N., Eyre, N. & Patterson, J. (2012). Effect of projectile design on coil gun performance. Franklin W. Olin College of Engineering.

Kaye, R. J. (2005). Operational requirements and issues for electromagnetic launchers. IEEE Transactions on Magnetics, 41(1), 194-199. doi: https://doi.org/10.1109/ELT.2004.1398047

Khandekar, S. (2016). Single-stage reluctance type coilgun. International Journal of Recent Research in Electrical and Electronics Engineering (IJRREEE), 3(3), 31-36.

Klimas, M., Grabowski, D. & Piaskowy, A. (2016). Efficiency analysis of an electromagnetic launcher. Elektryka, Silesian University of Technology Publishing, Poland.

McNab, I. R. (1999). Early electric gun research. IEEE Transactions on magnetics, 35(1), 250-261. doi: https://doi.org/10.1109/20.738413 De Medeiros, L. H, Reyne, G. & Meunier, G. (1998). Comparison of global force calculations on permanent magnets. IEEE Transactions on Magnetics, 34(5), 3560-3563. doi: https://doi.org/10.1109/20.717840 Meeker, D. (2004). Finite element method magnetics: version 4.2 user’s manual. Retrieved from http://www.femm.info/Archives/doc/manual.pdf. Rashid, M. H. (2011). Power electronics handbook: devices, circuits, and applications. 3rd ed. USA: Elsevier. Slade, W. G. (2005). A simple unified physical model for a reluctance accelerator. IEEE Transactions on Magnetics, 41(11), 4270-4276. doi: https://doi.org/10.1109/TMAG.2005.856320 Slade, W. G. (2006). Fast finite-element solver for a reluctance mass accelerator. IEEE Transactions on

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